First Fundamental Theorem of Calculus

Fundamental Theorem of Calculus (FTC 1)
If f(x) is ?continuous and F?(x) = f(x), then b
f(x)dx = F(b) ? F(a)
a
Notation: F(x)
b
= F(x)
x=b
= F(b) ? F(a)
a x=a
? x3 b x3 b b3 a3 Example 1. F(x) = F 2; x2dx = ?(x) = x =
3 ? 3 3 , 3 a a
Example 2. Area under one hump of sin x (See Figure 1.)
? ?
0
sin x dx = ?cos x
?
= ?cos ? ? (?cos 0) = ?(?1) ? (?1) = 2
0
1
??
Figure 1: Graph of f(x) = sin x for 0 ? x ? ?.
? 1 6 1
=
1 1
6 ? 0 =
6
5dx = x Example 3. x 6 0 0
1
???
Lecture 19 18.01 Fall 2006
Intuitive Interpretation of FTC:
dx
x(t) is a position; v(t) = x?(t) = is the speed or rate of change of x.
dt
? b
v(t)dt = x(b) ? x(a) (FTC 1)
a
R.H.S. is how far x(t) went from time t = a to time t = b (difference between two odometer readings).
L.H.S. represents speedometer readings.
n
i=1
x(b) ? x(a) = v(t) cancel each other, whereas an
?
v(ti)?t approximates the sum of distances traveled over times ?t
The approximation above is accurate if v(t) is close to v(t ) on the ith interval. The interpretation i
of x(t) as an odometer reading is no longer valid if v changes sign. Imagine a round trip so that
0. Then the positive and negative velocities
odometer would measure the total distance not the net distance traveled.